Research Article An Extensive Photometric Investigation of...

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Research Article An Extensive Photometric Investigation of the W UMa System DK Cyg M. M. Elkhateeb, 1,2 M. I. Nouh, 1,2 E. Elkholy, 1,2 and B. Korany 1,3 1 Astronomy Department, National Research Institute of Astronomy and Geophysics, Helwan, Cairo 11421, Egypt 2 Physics Department, College of Science, Northern Border University, Arar 1321, Saudi Arabia 3 Physics Department, Faculty of Applied Science, Umm Al-Qura University, Makkah 715, Saudi Arabia Correspondence should be addressed to M. I. Nouh; abdo [email protected] Received 25 August 2014; Revised 14 December 2014; Accepted 17 December 2014 Academic Editor: eodor Pribulla Copyright © 2015 M. M. Elkhateeb et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. DK Cyg ( = 0.4707) is a contact binary system that undergoes complete eclipses. All the published photoelectric data have been collected and utilized to reexamine and update the period behavior of the system. A significant period increase with rate of 12.590 × 10 −11 days/cycle was calculated. New period and ephemeris have been calculated for the system. A long term photometric solution study was performed and a light curve elements were calculated. We investigated the evolutionary status of the system using theoretical evolutionary models. 1. Introduction e eclipsing binary DK Cyg (BD +330 4304, 10.37–10.93 m v ) is a well known contact binary system with a period of about 0.4707 days. It was discovered as variable star earlier by Guth- nick and Prager [1], so their epoch of intensive observations is very long. e earliest photographic light curve classified the system as a W Ursae Majoris type. Rucinski and Lu [2] carried out the first spectroscopic observations and estimated the mass ratio as = 0.325 and classified the system as an A-subtype contact system with spectral type of A8V. Visual light curves were published by Piotrowski [3] and Tsesevitch [4] from Klepikova’s observations. First photoelectric observations for the system were car- ried out by Hinderer [5], while Binnendijk [6] observed the system photoelectrically in B- and V-bands and derived least squares orbital solution. e system DK Cyg was classified in the General Catalogue of Variable Stars as A7V [7], while Binnendijk [6] adopted it as A2. Mochnacki and Doughty [8] showed that the color index of the system judged its spectral type and found that the spectral type of the system is more likely to be about F0 to F2. Because the system DK Cyg is a summer object in the Northern hemisphere with 11.5-hour period and short durations of night, it is bound to remain ill- observed [9]. Only four complete light curves by Binnendijk [6], Paparo et al. [10], Awadalla [9], and Baran et al. [11] are published. Photoelectric observations and new times of minima have been carried out by many authors: Borkovits et al. [12], Sarounov´ a and Wolf [13], Drozdz and Ogloza [14], ubscher et al. [15], Dogru et al. [16], Hubscher et al. [17], Hubscher et al. [18], Erkan et al. [19], Diethelm [20], Dogru et al. [21], Simmons [22], Diethelm [23], and Diethelm [24]. In the present paper we are going to perform comprehen- sive photometric study for the system DK Cyg. e structures of the paper are as follows: Section 2 deals with the period change, Section 3 is devoted to the light curve modeling, and Section 4 presents the discussion and conclusion reached. 2. Period Change Although the period variation of contact binary systems of the W UMa-type is a controversial issue of binary star astrophysics, the cause of the variations (long as well as short term) is still a mystery for a discussion of possible physical mechanisms [25]. Magnetic activity cycle is one of the main mechanisms that caused a period variation together with Hindawi Publishing Corporation Journal of Astrophysics Volume 2015, Article ID 590673, 8 pages http://dx.doi.org/10.1155/2015/590673

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Research ArticleAn Extensive Photometric Investigation ofthe W UMa System DK Cyg

M. M. Elkhateeb,1,2 M. I. Nouh,1,2 E. Elkholy,1,2 and B. Korany1,3

1Astronomy Department, National Research Institute of Astronomy and Geophysics, Helwan, Cairo 11421, Egypt2Physics Department, College of Science, Northern Border University, Arar 1321, Saudi Arabia3Physics Department, Faculty of Applied Science, Umm Al-Qura University, Makkah 715, Saudi Arabia

Correspondence should be addressed to M. I. Nouh; abdo [email protected]

Received 25 August 2014; Revised 14 December 2014; Accepted 17 December 2014

Academic Editor: Theodor Pribulla

Copyright © 2015 M. M. Elkhateeb et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

DK Cyg (𝑃 = 0.4707) is a contact binary system that undergoes complete eclipses. All the published photoelectric data havebeen collected and utilized to reexamine and update the period behavior of the system. A significant period increase with rate of12.590 × 10−11 days/cycle was calculated. New period and ephemeris have been calculated for the system. A long term photometricsolution study was performed and a light curve elements were calculated. We investigated the evolutionary status of the systemusing theoretical evolutionary models.

1. Introduction

The eclipsing binary DK Cyg (BD +330 4304, 10.37–10.93mv)is a well known contact binary system with a period of about0.4707 days. It was discovered as variable star earlier by Guth-nick and Prager [1], so their epoch of intensive observationsis very long. The earliest photographic light curve classifiedthe system as a W Ursae Majoris type. Rucinski and Lu [2]carried out the first spectroscopic observations and estimatedthe mass ratio as 𝑞 = 0.325 and classified the system as anA-subtype contact system with spectral type of A8V. Visuallight curves were published by Piotrowski [3] and Tsesevitch[4] from Klepikova’s observations.

First photoelectric observations for the system were car-ried out by Hinderer [5], while Binnendijk [6] observed thesystem photoelectrically in B- and V-bands and derived leastsquares orbital solution. The system DK Cyg was classifiedin the General Catalogue of Variable Stars as A7V [7], whileBinnendijk [6] adopted it as A2. Mochnacki and Doughty [8]showed that the color index of the system judged its spectraltype and found that the spectral type of the system is morelikely to be about F0 to F2. Because the system DK Cyg isa summer object in the Northern hemisphere with 11.5-hour

period and short durations of night, it is bound to remain ill-observed [9]. Only four complete light curves by Binnendijk[6], Paparo et al. [10], Awadalla [9], and Baran et al. [11]are published. Photoelectric observations and new times ofminima have been carried out by many authors: Borkovitset al. [12], Sarounova and Wolf [13], Drozdz and Ogloza [14],Hubscher et al. [15], Dogru et al. [16], Hubscher et al. [17],Hubscher et al. [18], Erkan et al. [19], Diethelm [20], Dogruet al. [21], Simmons [22], Diethelm [23], and Diethelm [24].

In the present paper we are going to perform comprehen-sive photometric study for the systemDKCyg.The structuresof the paper are as follows: Section 2 deals with the periodchange, Section 3 is devoted to the light curve modeling, andSection 4 presents the discussion and conclusion reached.

2. Period Change

Although the period variation of contact binary systemsof the W UMa-type is a controversial issue of binary starastrophysics, the cause of the variations (long as well as shortterm) is still a mystery for a discussion of possible physicalmechanisms [25]. Magnetic activity cycle is one of the mainmechanisms that caused a period variation together with

Hindawi Publishing CorporationJournal of AstrophysicsVolume 2015, Article ID 590673, 8 pageshttp://dx.doi.org/10.1155/2015/590673

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the mass exchange between the components of each system.Kaszas et al. [26] stated that the long term period variationmay be interpreted by a perturbation of the third companionor surface activity of the system components.

Observations by Binnendijk [6] showed a change of thesecondary minimum depth and a new linear light elementwas derived. Period study by Paparo et al. [10] showed thatthe orbital period of the systemDKCyg increases and the firstparabolic light elements were calculated, which confirmedthe light curve variability. Kiss et al. [25] updated the linearephemeris of DK Cyg, while Awadalla [9] recalculated a newquadratic element for the system and confirmed the lightcurve variability suggested by Paparo et al. [10]. Wolf et al.[27] used a set of 101 published times of minimum coveringthe interval between 1926 and 2000 in order to update thequadratic element calculated by Awadalla [9]. They showedthat the period increases by the rate 11.5 × 10−11 days/cycle.Borkovits et al. [28] follow the period behavior of the systemusing set of published minima from HJD 2424760 to HJD2453302.

In this paper we studied the orbital period behavior of thesystem DK Cyg using the (𝑂 − 𝐶) diagram based on morecomplete data set collected from the literatures and databasesof BAV, AAVSO, and BBSAG observers. Part of our collecteddata set was given by Kreiner et al. [29]; unpublished Hippar-cos observations and main part were downloaded from web-site (http://astro.sci.muni.cz/variables/ocgate/); Table 1 listedonly those minima not listed on the mentioned websites. Atotal of 195 minima times were incorporated in our analysiscovering about 86 years (66689 orbital revolutions) from1927 to 2013. It is clear that our set of data added about94 of new minima and increases the interval limit of theorbital period study about 13 years more than the data ofWolf et al. [27], which may give more accurate insight onthe period behavior of the system. The different types of thecollected minima (i.e., photographic, visual, photoelectric,andCCD)wereweighted according to their type.The residual(𝑂 − 𝐶)’s were computed using Binnendijk [6] ephemeris (1)and represented in Figure 1. No distinction has been madebetween primary and secondary minima:

Min 𝐼 = 2437999.5838 + 0.47069055 ∗ 𝐸. (1)

It can be seen from the figure that the behavior of the orbitalperiod of the system DK Cyg shows a parabolic distributionwhich is generally interpreted by the transfer of mass fromone component to the other of binary. Reasonable linear leastsquares fit of the data available improved the light elementsgiven in (1) to

Min 𝐼 = 2437999.5961±0.0192

+ 0𝑑

.47069206 ∗ 𝐸±0.0243

. (2)

The linear element yields a new period of 𝑃 = 0𝑑.47069206days which is longer by 0.13 seconds with respect to the valuegiven by Binnendijk [6]. Quadratic least squares fit gives

Min 𝐼 = 2437999.5803±0.0192

+ 0𝑑

.47069064 ∗ 𝐸±0.0237

+ 6.284 × 10−11

× 𝐸2

±2.5654×10

−12

.(3)

−40

000

−20

000 0

2000

0

4000

0

−30

000

−10

000

1000

0

3000

0

5000

0

Integer cycle (E)

−0.04

0

0.04

0.08

0.12

−0.02

0.02

0.06

0.1

0.14

(O−C)

Figure 1: Period behavior of DK Cyg.

0

0.04

0.02

−40

000

−20

000 0

2000

0

4000

0

−30

000

−10

000

1000

0

3000

0

5000

0

Integer cycle (E)

−0.02

−0.04

−0.06

(O−C)q

Figure 2: Calculated residuals from the quadratic ephemeris.

The rate of period increasing resulting from the quadraticelements (3) is 𝑑𝑃/𝑑𝐸 = 12.568 × 10−11 days/cycle or9.746 × 10−8 days/year or 0.84 seconds/century. More futuresystematic and continuous photometric observations areneeded to follow a continuous change in the orbital period ofthe systemDKCyg which may show a periodic behavior.Thefourth column of Table 1 represents the quadratic residuals(𝑂 − 𝐶)𝑞 calculated using the new element of (2) andrepresented in Figure 2. All published linear and quadraticelements together with that resulting from our calculationsare listed in Table 2. It is noted from the table that thequadratic term resulting from our calculations has slightlyhigher value than that calculated by Awadalla [9], Wolf et al.[27], and Borkovits et al. [28]. This can be interpreted by theincreasing of the set of minima in our study compared to theone they used (nearly double); also we covered an intervallarger than the one they used.

3. Light Curve Modeling

Light curve modeling for the system DK Cyg by Mochnackiand Doughty [8] using Binnendijk [6] observations in V-band showed nonmatching between the theoretical curve and

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Table 1: Times of minimum light for DK Cyg.

HJD Method 𝐸 (𝑂 − 𝐶) (𝑂 − 𝐶)𝑞 References HJD Method 𝐸 (𝑂 − 𝐶) (𝑂 − 𝐶)𝑞 References2434179.4690 Vis −8116 0.00970 0.00971 1 2451749.4450 pe 29212 0.04885 −0.00370 32447758.4352 Pe 20733 0.02423 −0.00099 2 2451777.6740 ccd 29272 0.03642 −0.01636 52447790.4437 Pe 20801 0.02577 0.00037 2 2452163.6590 ccd 30092 0.05517 −0.00074 52447963.6620 Pe 21169 0.02995 0.00354 3 2452245.5592 ccd 30266 0.05521 −0.00137 52447963.8960 Pe 21169.5 0.02860 0.00220 3 2452253.5613 ccd 30283 0.05557 −0.00108 52448265.1380 Pe 21809.5 0.02865 0.00046 4 2452441.8384 ccd 30683 0.05645 −0.00176 52448265.1382 Pe 21809.5 0.02885 0.00066 4 2452512.4415 pe 30833 0.05597 −0.00284 72448272.1987 Pe 21824.5 0.02899 0.00076 4 2452525.6231 ccd 30861 0.05824 −0.00069 52448297.6160 Pe 21878.5 0.02900 0.00062 3 2452526.5644 ccd 30863 0.05816 −0.00078 52448302.7930 Pe 21889.5 0.02841 −0.00001 3 2452811.8062 ccd 31469 0.06148 0.00013 52448308.2078 Pe 21901 0.03027 0.00182 4 2453223.4286 ccd 32343.5 0.06500 −0.00006 82448308.2079 Pe 21901 0.03037 0.00192 4 2453228.3681 ccd 32354 0.06225 −0.00274 82448336.4491 Pe 21961 0.03013 0.00152 4 2453246.4950 ccd 32392.5 0.06756 0.00242 82449988.5840 ccd 25471 0.04120 0.00182 5 2453247.4346 ccd 32394.5 0.06578 0.00063 82450003.6456 ccd 25503 0.04070 0.00122 5 2453285.3260 ccd 32475 0.06659 0.00110 82450313.8240 ccd 26162 0.03403 −0.00765 5 2453286.2657 ccd 32477 0.06491 −0.00059 82450341.6130 ccd 26221 0.05229 0.01041 5 2453302.2672 ccd 32511 0.06293 −0.00271 82450397.6060 ccd 26340 0.03311 −0.00917 5 2454799.5505 ccd 35692 0.07959 0.00004 92450692.7400 ccd 26967 0.04414 −0.00030 5 2455043.8381 ccd 36211 0.07882 −0.00310 102451000.0990 pe 27620 0.04221 −0.00452 5 2455062.6680 ccd 36251 0.08107 −0.00105 112451095.6600 ccd 27823 0.05303 0.00557 5 2455088.5544 ccd 36306 0.07953 −0.00285 102451160.5980 ccd 27961 0.03573 −0.01222 5 2455810.6029 ccd 37840 0.08870 −0.00096 122451379.4820 pe 28426 0.04863 −0.00101 3(1) Szafraniec [30]; (2) Hubscher et al. [31]; (3) Wolf et al. [27]; (4) Hipparcos observations (unpublished); (5) Baldwin and Samolyk [32]; (6) Kiss et al. [25]; (7)Borkovits et al. [33]; (8) Borkovits et al. [28]; (9) Gerner [34]; (10) Menzies [35]; (11) Samolyk [36]; (12) Simmons [22].

Table 2: The light elements of DK Cyg.

JD Period Quadratic term References2437999.5838 0.470690550 Binnendijk [6]2437999.5828 0.470690660 5.390 × 10−10 Paparo et al. [10]2437999.5825 0.470690730 5.760 × 10−11 Awadalla [9]2451000.0999 0.470692900 Kiss et al. [25]2437999.5825 0.470690640 5.750 × 10−11 Wolf et al. [27]2451000.1031 0.470693909 5.862 × 10−11 Borkovits et al. [28]2437999.5961 0.470692060 Present work2437999.5803 0.470690640 6.284 × 10−11 Present work

the observations.The photometricmass ratio calculated fromtheir acceptedmodel was 𝑞ph = 0.33±0.02, while the spectro-scopic value estimated using radial velocity study by Rucinskiand Lu [2] is 𝑞sp = 0.325 ± 0.04. Baran et al. [11] estimatedan alternating model of spectroscopic and photometric databased on iterative solutions.Their model shows a better fit byintroducing a cool spot on the surface of the more luminouscomponent and adopted the third light as free parameter inthe computations. On the other hand Rucinski and Lu [2]stated that they did not find any evidence for the existenceof a third component in the system during their spectro-scopic study. They refer the probability for the presence of

a third star in the system to the (𝑂 − 𝐶) diagram [29], whichshows sinusoidal variation. This evidence is weak becauseonly one cycle is covered up to date.

In the present work we used complete light curvespublished by Binnendijk [6], Paparo et al. [10], Awadalla[9], and Baran et al. [11] in V-band through a long termphotometric solution study in order to estimate the physicalparameters of the system and to follow its evolutionary status.The collected light curves showed a flat-bottom minima andO’Connell effect. Observations by Paparo et al. [10] andAwadalla [9] displayed some scattering specially at the twomaxima of their light curves. Also the observed light curve

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Table 3: Photometric solutions for DK Cyg.

Parameter Binnendijk [6] Paparo et al. [10] Awadalla [9] Baran et al. [11]𝐴 5500 5500 5500 5500𝑖 (∘) 80.59 ± 0.12 80.22 ± 0.21 80.83 ± 0.27 79.97 ± 0.06𝑔1

= 𝑔2

0.32 0.32 0.32 0.32𝐴1

= 𝐴2

0.5 0.5 0.5 0.5𝑞 (𝑀2

/𝑀1

) 0.306∗ 0.306∗ 0.306∗ 0.306∗

Ω1

=Ω2

2.4064 ± 0.002 2.4077 ± 0.005 2.3325 ± 0.004 2.3886 ± 0.001Ωin 2.4794 2.4794 2.4794 2.4794Ωout 2.2888 2.2888 2.2888 2.2888𝑇1

(∘K) 7500∗ 7500∗ 7500∗ 7500∗

𝑇2

(∘K) 6767 ± 4 6726 ± 7 6726 ± 9 6759 ± 2𝑟1

pole 0.4696 ± 0.0007 0.4694 ± 0.0013 0.4861 ± 0.0014 0.4735 ± 0.0003𝑟1

side 0.5091 ± 0.0010 0.5087 ± 0.0018 0.5328 ± 0.0021 0.5145 ± 0.0004𝑟1

back 0.5400 ± 0.0013 0.5395 ± 0.0024 0.5716 ± 0.0029 0.5471 ± 0.0005𝑟2

pole 0.2792 ± 0.0008 0.2789 ± 0.0014 0.2980 ± 0.0017 0.2835 ± 0.0003𝑟2

side 0.2932 ± 0.0010 0.2929 ± 0.0017 0.3166 ± 0.0022 0.2985 ± 0.0004𝑟2

back 0.3414 ± 0.0019 0.3407 ± 0.0034 0.3982 ± 0.0068 0.3522 ± 0.0008Spot A of star 1

Colatitude 130∗ 130∗ 130∗ 130∗

Longitude 180∗ 180∗ 180∗ 180∗

Spot radius 33.61 ± 0.230 30.74 ± 0.437 27.23 ± 1.14 35.014 ± 0.09Temp. factor 0.796 ± 0.003 0.840 ± 0.007 0.924 ± 0.01 0.819 ± 0.001

Spot A of star 2Colatitude 120∗ 120∗ 120∗ 120∗

Longitude 290∗ 290∗ 290∗ 290∗

Spot radius 32.99 ± 3.60 29.42 ± 1.34 33.08 ± 1.24 29.44 ± 1.20Temp. factor 1.01 ± 0.01 1.01 ± 0.01 1.17 ± 0.01 1.02 ± 0.002∑(𝑂 − 𝐶)

2 0.0229 0.02909 0.02458 0.0453∗Not adjusted.

by Awadalla [9] shows sudden increase in the light levelat secondary minimum with respect to the other collectedcurves.

Photometric analysis for the studied light curves of thesystemDKCygwas carried out usingMode 3 (overcontact) ofWDint56a Package [43] based on the 2009 version of Wilsonand Devinney (W-D) code with Kurucz model atmospheres[44–46]. The observed light curves were analyzed usingall individual observations. Appropriate gravity darkeningand bolometric albedo exponents were assumed for theconvective envelope. We adopted g1 = g2 = 0.32 [47] andA1 = A2 = 0.5 [48]. Bolometric limb darkening values areadopted using the table of van Hamme [49]. Temperature ofthe primary star was adopted according to Baran et al. [11]model (T1 = 7500∘K).

The adjustable parameters are the mean temperature ofthe secondary component T2, orbital inclination i, and thepotential of the two components Ω = Ω1 = Ω2, while thespectroscopic mass ratio (qsp = 0.306) by Baran et al. [11] wasfixed for all calculatedmodels together with the primary star’stemperature (T1).

We startedmodeling using as initial values the parametersof Baran et al. [11] solution based on cool spot on the luminous

components and a third light as a free parameter. The usedparameters show disagreement between the theoretical andobserved light curves, except for Baran et al. observations.Regarding the conclusion of Rucinski andLu [2], which stateda weak evidence of presence of a third component, we triedto construct a spotted model without the third light. Weconstructed a model including two spots; the first one is acool spot located on surface of the more massive component,while the other is a hot spot located on the surface of the othercomponent. The accepted model reveals good agreementbetween theoretical and observed light curves for all collecteddata. Table 3 lists the calculated parameters for the fourlight curves, while Figure 3 represented the theoretical lightcurves according to the accepted solution together with thereflected points in V-band. The ∑(𝑂 − 𝐶)2 values in Table 3are indicative of comparisons in future studies, since thenumber of observations and the accuracy are not the samein the four light curves. Absolute physical parameters foreach component of the systemDKCyg were calculated basedon the results of the radial velocity data of Baran et al. [11]and our new photometric solution for each light curve. Thecalculated parameters are listed in Table 3. The results showthat the primary component is more massive and hotter than

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Table 4: Absolute physical parameters for DK Cyg.

Parameter Binnendijk [6] Paparo et al. [10] Awadalla [9] Baran et al. [11]𝑀1⊙

1.7363 ± 0.0709 1.7358 ± 0.0709 1.7679 ± 0.0722 1.7438 ± 0.0712𝑀2⊙

0.5313 ± 0.0217 0.5312 ± 0.0217 0.5410 ± 0.0221 0.5336 ± 0.0218𝑅1⊙

1.7037 ± 0.0696 1.7029 ± 0.0695 1.7635 ± 0.0720 1.7178 ± 0.0701𝑅2⊙

1.0129 ± 0.0414 1.0118 ± 0.0413 1.0811 ± 0.0441 1.0285 ± 0.0420𝑇1⊙

1.2980 ± 0.0530 1.2980 ± 0.0530 1.2980 ± 0.0530 1.2980 ± 0.053𝑇2⊙

1.1712 ± 0.0478 1.1641 ± 0.0475 1.1641 ± 0.0475 1.1698 ± 0.0478𝑀1 bol 2.4617 ± 0.1005 2.4627 ± 0.1005 2.3868 ± 0.0974 2.4438 ± 0.1000𝑀2 bol 4.0375 ± 0.1648 4.0662 ± 0.1660 3.9224 ± 0.1601 4.0094 ± 0.1637𝐿1⊙

8.2285 ± 0.3359 8.2208 ± 0.3356 8.8163 ± 0.3600 8.3653 ± 0.3415𝐿2⊙

1.9276 ± 0.0787 1.8772 ± 0.0766 2.1431 ± 0.0875 1.9780 ± 0.0808Note: subscripts 1 and 2 mean primary and secondary component, respectively.

Table 5: Physical parameters of the five A-type contact binaries.

Star name Parameters References𝑀1

(𝑀⊙

) 𝑀2

(𝑀⊙

) 𝑅1

(𝑅⊙

) 𝑅2

(𝑅⊙

) 𝐿1

(𝐿⊙

) 𝐿2

(𝐿⊙

)

YY CrB 1.404 0.339 1.427 0.757 2.58 6.68 1AW UMa 1.6 0.121 1.786 0.739 7.47 0.804 2EQ Tau 1.214 0.541 1.136 0.787 1.31 0.6 3RR Cen 1.82 0.38 2.1 1.05 8.89 2.2 4V566 Oph 1.41 0.34 1.45 0.77 4.46 1.23 5(1) Essam et al. [38], (2) Elkhateeb and Nouh [39], (3) Elkhateeb and Nouh [40], (4) Yang et al. [41], and (5) Degirmenci [42].

0 0.4 0.8 1.20.2 0.6 1Phase

0.8

1.2

1.6

0.6

1

1.4

BinPap

Awa

Bar

−0.2

Flux

V

Figure 3: Observed and synthetic light curves of Binndijik [6] (Bin),Paparo et al. [10] (Pap), Awadalla [9] (Awa), and Baran et al. [11](Bar), for the system DK Cyg.

the secondary component. A three-dimensional geometricalstructure for the systemDKCyg is displayed in Figure 4 usingthe software Package Binary Maker 3.03 [50] based on thecalculated parameters resulting from our models.

4. Discussion and Conclusion

Studying of the period behavior of the system DK Cygbased on all available published times of minima, covering86 yr of observations including 195 times of light minima,shows a continuous period increase with the rate 𝑑𝑃/𝑑𝐸 =12.590 × 10

−11 days/cycle or 9.763 × 10−8 days/year or 0.84

seconds/century. New linear and quadratic elements werecalculated using all available published data and yield a newperiod of 𝑃 = 0.47069203 days. A long term photomet-ric study was performed using published observations byBinnendijk [6], Paparo et al. [10], Awadalla [9], and Baranet al. [11]. More systematic and continuous photometricobservations for the system DK Cyg are needed to confirma continuous change in the period and follow its light curvevariation.

One of the difficulties forWUMa binaries is to use stellarmodels of single stars to investigate the evolutionary status ofthese systems. However, using these theoretical models maygive approximate view about the evolutionary status of thesystem.

We used the physical parameters listed in Table 4 toinvestigate the current evolutionary status of DK Cyg. InFigures 5 and 6, we plotted the components of DK Cygon onthe mass-luminosity (𝑀-𝐿) andmass-radius (𝑀-𝑅) relationsalong with the evolutionary tracks computed by Girardi et al.[51] for both zero age main sequence stars (ZAMS) and ter-minal age main sequence stars (TAMS) with metallicity 𝑧 =0.019. As it is clear from the figures, the primary componentof the system is located nearly on the ZAMS for both the𝑀-𝐿 and𝑀-𝑅 relations. The secondary component is above theTAMS track for 𝑀-𝐿 and the 𝑀-𝑅 relations. For the sakeof comparison, we plotted sample of A-type contact binarieslisted in Table 5. The components of DK Cyg have the samebehavior of the selected A-type systems.

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6 Journal of Astrophysics

Binnendijk (1964) Paparo et al. (1985)

Awadalla (1994) Baran et al. (2004) Phase = 0.64

Phase = 0.64 Phase = 0.64

Phase = 0.64

Figure 4: Three-dimensional models of the components of DK Cyg.

−1 0 0.5 1

−0.75

−0.25

0.25

0.75

1.25

ZAMS

TAMS

log M/M⊙

−0.5

log R

/R⊙

Figure 5: The position of the components of DK Cyg on the mass-radius diagram. The filled symbols denote the primary componentand the open symbols represent the secondary component. The starsymbols denote the sample of the selected A-type systems listed inTable 5.

The mass-effective temperature relation (𝑀-𝑇eff) forintermediate and low mass stars [37] is displayed in Figure 7.The location of our mass and radius on the diagram revealeda good fit for the primary and poor fit for the secondarycomponents. This gave the same behavior of the system onthe mass-luminosity and mass-radius relations.

0 0.5 1

ZAMS

TAMS

0

2

4

1

3

5

log M/M⊙

log L

/L⊙

−2

−3

−4

−1

−1 −0.5

Figure 6: The position of the components of DK Cyg on the mass-luminosity diagram.The filled symbols denote the primary compo-nent and the open symbols represent the secondary component.Thestar symbols denote the sample of the selected A-type systems listedin Table 5.

Conflict of Interests

The authors declare that they have no conflict of interestsregarding the publication of this paper.

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Journal of Astrophysics 7

0 1 20.5 1.5

0

0.4

0.8

−1 −0.5−1.5−0.4

log M/M⊙

log T

/T⊙

Figure 7: Position of the components of DK Cyg on the empiricalmass-𝑇eff relation for low-intermediate mass stars by Malkov [37].The filled symbols denote the primary component and the opensymbols represent the secondary component.

Acknowledgments

This research has made use of NASA’s ADS and the availableon-line material of the IBVS. The authors sincerely thank Dr.Bob Nelson, who allowed them to use his windows interfacecode WDwint56a, for his helpful discussions and advice.

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